Partition identities and complete assembly

If raw ambient sectors are used instead of the main post-adjoint convention, define

Then

All spatial derivatives of the cutoff generated by the vertex and adjoints are retained. If the partition sums to one near $\mathrm{supp}\zeta$, its summed motion terms cancel. For the cutoffs in (eq:partition),

For raw sectors defined after an adjoint, ${\text{I}}_e({\vartheta }_{e,j,\delta }{\mathcal{L}}_{e,c_{\delta }})$, differentiation also produces ${\text{I}}_e((D_{\delta }{\vartheta }_{e,j,\delta }){\mathcal{L}}_{e,c_{\delta }})$. Moving a cutoff through an adjoint generates additional spatial commutator terms. These conventions are not termwise interchangeable.

If the chosen exact differentiated partition gives a finite sum $v_{j,\delta }$, and each term has a verified finite power–logarithm expansion with an integrable remainder, then the complete coefficients are the sums of those coefficients. The remainder is bounded by the sum of the majorants and the maximum of the finite test orders. This is finite addition, not a sector estimate supplied by the partition.